When does $e^{-\lvert \tau\rvert }$ maximize Fourier extension for a conic section?

TL;DR

This paper surveys recent advances in sharp restriction theory, focusing on when the exponential function maximizes Fourier extension for conic sections, with results for spherical, hyperbolic, and cone cases.

Contribution

It provides a comprehensive overview of recent results and techniques related to maximizers of Fourier extension inequalities for conic sections, including algebraic and transform methods.

Findings

Maximizers identified for spherical and hyperbolic cases.

Negative results for the cone case using Penrose transform.

Clarification of conditions under which exponential functions maximize Fourier extension.

Abstract

In the past decade, much effort has gone into understanding maximizers for Fourier restriction and extension inequalities. Nearly all of the cases in which maximizers for inequalities involving the restriction or extension operator have been successfully identified can be seen as partial answers to the question in the title. In this survey, we focus on recent developments in sharp restriction theory relevant to this question. We present results in the algebraic case for spherical and hyperbolic extension inequalities. We also discuss the use of the Penrose transform leading to some negative answers in the case of the cone.